Sintomatología

desired responses are influenced by several factors and levels. Compared to a conventional experimental approach when only one-factor-at-a-time (OFAT) is varied, DoE has several benefits. It can be used to analyse factor interactions [33, 157] which is impossible with OFAT, to estimate error variation in the experiment, and to economically plot response contours [157]. In nanocomposites studies, some popular designs are factorial design [33,

158], the Taguchi method [36, 38, 65, 159, 160], the Central Composite Design (CCD) [39, 40, 161, 162], and Box-Behnken design (BBD) [37, 83, 163, 164].

The factorial design can be full or fractioned design depending whether all factors and all experiment levels are needed [157]. It can be used to develop an empirical model of the response variables in terms of the factors. The limitation of this method is it has no repeat experiments, so it is unable to estimate error variation or model adequacy.

The Taguchi method is a fractional factorial that offers simpler design than conventional factorial design. It is commonly applied to detect main effects. It has been used to observe the main processing factors in polyamide microcellular nanocomposite preparation [160], the main effects in thermoplastic vulcanizate nanocomposite preparation [159], and the effect of material content [38, 65]. Because it was developed from a fractioned factorial, the efficiency of the design, in term of runs, is very good. However, its limitation is that it’s not possible to observe curvature area.

In contrast, response surface methodology (RSM) is a good tool for analysing results and predicting the optimum conditions from a developed surface. Several studies have used CCD, (a type of RSM) to optimise additive levels in nanocomposites [39, 40, 161]. In this technique, the number of experiments is calculated by 2f + 2f+ nc. where f and nc are the

number of design factors and repeated runs. The benefit of CCD is the availability of axial points (shown in Figure 2.5(a)), which expand the boundaries. As a consequence, however, there are more experimental runs, and some corner points are extreme experiment conditions. The drawback of this design is that it is rather expensive, and some points may not be possible to produce.

(a) (b)

Another RSM that is commonly used is the Box-Behnken design (BBD). It was developed from an incomplete 3kfactorial. The BBD is shown inFigure 2.5(b). Some studies have used it to optimise processing conditions [37, 83, 163, 164]. The benefits of the BBD are that it is rotatable and more efficient due to the absence of axial points and extreme design points [157]. However, for experiments with more than three factors, the number of runs might be more than CCD or fractioned factorial design. The comparison on number of runs for BBD and CCD is shown inTable 2.7.

Table 2.7 Available response surface design (with number of runs) [165]

Design Number of Factors

2 3 4 5 6 7 8 9 10 CCD Full 13-14 20 30-31 52-54 90 152-160 CCD half 32-33 53-54 88-90 154-160 CCD quarter 90 156-160 CCD eighth 158-160 BBD 15 27 46 54 66 130 170

In term of model validation, it is assumed that model errors should follow a normal distribution. Then should be no non-constant variance, and errors should be independent [157, 165, 166]. Another factor to be considered at validation is outlier detection, which indicates either measurement error or unique characteristics in a particular sample population. Model adequacy can be analysed by various statistical measurements, such as the estimated error standard deviation, the coefficient of determination (R2), or lack of fitness test [157]. Estimated error standard deviation reflects the closeness of the predicted value to the actual response. The lower the value of this parameter, the closer the predicted value to the actual response. The value is calculated according toEquation 2.2.

Equation 2.2

Where :

= Error mean of squares = = Error sum of squares

The coefficient of determination (R2) is also a measurement assessing model goodness of fit. However, this measurement is not suitable if model comparison is done because it does not adjust the number of predictors and observations. Instead of using R2, R2adjusted is more appropriate to compare the number of models because it considers the number of calculated variables [157, 165-167]. The equations for calculating R2 and R2adjusted are shown in

Equations 2.3and2.4.

Equation 2.3 Equation 2.4

where :

= Total sum of squares (n – 1) = Total degree of freedom

Lack of fit can also be used to assess model adequacy when there are some repeated observations [157, 166]. The measurement concept is generated from the sum of squares error which is fractioned into lack of fit and pure error, respectively. However, it might yield an inaccurate analysis if some significant factors are not included in the model terms. In addition, the accuracy of analysis decreases if each sample has its own standard deviation calculated from a sub-set of specimens. These standard deviations will not be considered in the lack of fit calculation, because standard deviation in the lack of fit calculation is based on average values from each repeated samples, not specimens. This suggests that the calculation may not represent genuine variation in actual conditions. In addition, insignificant lack of fit does not always mean the model is correct, for example, in cases where there is only a small variation around the response [166, 167].

In the literature, various approaches have been adopted to justify model adequacy. One study found that significance of regression and lack of fit are useful methods for evaluating models [168]. Another study used normality testing and residual plots to justify model adequacy [169]. R2was also applied to check model goodness of fit in several studies [164, 170-172], even when lack of fit was significant [173, 174]. Model adequacy was inferred from signal-to-noise ratio, when comparing predicted response to its error [170].

While there is little consensus about analysis of model adequacy, R2 can usually be used as long as consideration is given to actual conditions [166].

2.7. Summary

In summary, nanofillers are a popular technique to improve polymer material properties with a small amount of filler loading. Formulation and process contribute to nanocomposite morphology and mechanical properties. Melt compounding is the most popular method to produce nanocomposites commercially. However, there is still no consensus on the optimum level of filler, type of filler, ratio of compatibiliser, type of equipment and optimum processing conditions in PE nanocomposites. More work is needed to enhance the understanding of this important research area.

A review of studies of nanocomposite characterization suggests there is no single technique to fully characterize nanocomposite morphology. Mechanical testing is commonly used to analyse bulk material properties and corroborate morphology characterization. A combination of XRD, TEM, and SEM produce the most convincing conclusions. XRD and TEM are a useful combination to characterize nanocomposite morphology, while XRD and SEM are a useful combination to characterize microcomposite morphology. Statistical analysis of micrographs is needed to draw more accurate conclusions.

Experimental design is a statistical approach to conducting experiments that is useful when there are interactions between factors. Some efficient designs like the Box-Behnken design saves time and money by reducing the number of observations while still considering all experiment factors. It can be used to optimise variables such as processing conditions, clay loading, and compatibiliser ratio, by analysing the effect of independent factors and interaction terms.